In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Appell–Humbert theorem describes the
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...
s on a
complex torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
or complex
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...
.
It was proved for 2-dimensional tori by and , and in general by
Statement
Suppose that
is a complex torus given by
where
is a lattice in a complex vector space
. If
is a
Hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear map, linear in each of its arguments, but a sesquilinear f ...
on
whose imaginary part
is integral on
, and
is a map from
to the unit circle
, called a semi-character, such that
:
then
:
is a 1-
cocycle
In mathematics a cocycle is a closed cochain (algebraic topology), cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in gr ...
of
defining a line bundle on
. For the trivial Hermitian form, this just reduces to a
character. Note that the space of character morphisms is isomorphic with a real torus
if
since any such character factors through
composed with the exponential map. That is, a character is a map of the form
for some covector
. The periodicity of
for a linear
gives the isomorphism of the character group with the real torus given above. In fact, this torus can be equipped with a complex structure, giving the
dual complex torus.
Explicitly, a line bundle on
may be constructed by
descent
Descent may refer to:
As a noun Genealogy and inheritance
* Common descent, concept in evolutionary biology
* Kinship, one of the major concepts of cultural anthropology
**Pedigree chart or family tree
**Ancestry
**Lineal descendant
**Heritage
** ...
from a line bundle on
(which is necessarily trivial) and a
descent data, namely a compatible collection of isomorphisms
, one for each
. Such isomorphisms may be presented as nonvanishing holomorphic functions on
, and for each
the expression above is a corresponding holomorphic function.
The Appell–Humbert theorem says that every line bundle on
can be constructed like this for a unique choice of
and
satisfying the conditions above.
Ample line bundles
Lefschetz proved that the line bundle
, associated to the Hermitian form
is ample if and only if
is positive definite, and in this case
is very ample. A consequence is that the complex torus is algebraic if and only if there is a positive definite Hermitian form whose imaginary part is integral on
See also
*
Complex torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
for a treatment of the theorem with examples
References
*
*
*
*
*
{{DEFAULTSORT:Appell-Humbert theorem
Abelian varieties
Theorems in algebraic geometry
Theorems in complex geometry